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section of a fiber bundle

Let $\funcdef{p}{E}{B}$ be a fiber bundle, denoted by $\xi.$

A section of $\xi$ is a continuous map $\funcdef{s}{B}{E}$ such that the composition $p\comp s$ equals the identity. That is, for every $b\in B,$ $s(b)$ is an element of the fiber over $b.$ More generally, given a topological subspace $A$ of $B,$ a section of $\xi$ over $A$ is a section of the restricted bundle $\funcdef{\restr{p}{A}}{\inv{p}(A)}{A}.$

The set of sections of $\xi$ over $A$ is often denoted by $\Gamma(A;\xi),$ or by $\Gamma(\xi)$ for sections defined on all of $B.$ Elements of $\Gamma(\xi)$ are sometimes called global sections, in contrast with the local sections $\Gamma(U;\xi)$ defined on an open set $U.$

Remark 1 If $E$ and $B$ have, for example, smooth structures, one can talk about smooth sections of the bundle. According to the context, the notation $\Gamma(\xi)$ often denotes smooth sections, or some other set of suitably restricted sections.

Example 1 If $\xi$ is a trivial fiber bundle with fiber $F,$ so that $E=F\cross B$ and $p$ is projection to $B,$ then sections of $\xi$ are in a natural bijective correspondence with continuous functions $\funcsig{B}{F}.$

Example 2 If $B$ is a smooth manifold and $E=TB$ its tangent bundle, a (smooth) section of this bundle is precisely a (smooth) tangent vector field.

In fact, any tensor field on a smooth manifold $M$ is a section of an appropriate vector bundle. For instance, a contravariant $k$ -tensor field is a section of the bundle $TM^{\otimes k}$ obtained by repeated tensor product from the tangent bundle, and similarly for covariant and mixed tensor fields.

Example 3 If $B$ is a smooth manifold which is smoothly embedded in a Riemannian manifold $M,$ we can let the fiber over $b\in B$ be the orthogonal complement in $T_b M$ of the tangent space $T_b B$ of $B$ at $b$ . These choices of fiber turn out to make up a vector bundle $\nu(B)$ over $B,$ called the normal bundle of $B$ . A section of $\nu(B)$ is a normal vector field on $B.$

Example 4 If $\xi$ is a vector bundle, the zero section is defined simply by $s(b)=0,$ the zero vector on the fiber.

It is interesting to ask if a vector bundle admits a section which is nowhere zero. The answer is yes, for example, in the case of a trivial vector bundle, but in general it depends on the topology of the spaces involved. A well-known case of this question is the hairy ball theorem, which says that there are no nonvanishing tangent vector fields on the sphere.

Example 5 If $\xi$ is a principal $G$ -bundle, the existence of any section is equivalent to the bundle being trivial.

Remark 2 The correspondence taking an open set $U$ in $B$ to $\Gamma(U;\xi)$ is an example of a sheaf on $B.$

section of a fiber bundle is owned by Antonio.

How to Cite This Entry

Antonio. "section of a fiber bundle" (version 7). PlanetMath.org. Freely available at http://planetmath.org/SectionOfAFiberBundle.html.

Classification

AMS MSC:

55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles)

Pending Errata and Addenda

None. [ View all 1 ]

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